A unified framework for pointwise convergence to the initial data of heat equations in metric measure spaces

TL;DR

This paper establishes a unified framework for ensuring pointwise convergence to initial data for heat equations across various metric measure spaces and operators, broadening understanding of heat kernel behavior.

Contribution

It introduces general conditions on heat kernels that guarantee pointwise convergence, applicable to a wide range of operators including fractional and perturbed Laplacians.

Findings

Conditions satisfied by diverse operators like fractional Laplacian and Dunkl Laplacian.

Characterization of convergence for Laplace operator with Hardy potential.

Extension of results to nonhomogeneous equations and nonlinearities.

Abstract

Given a metric measure space $(\mathcal{X}, d, \mu)$ satisfying the volume doubling condition, we consider a semigroup $\{S_t\}$ and the associated heat operator. We propose general conditions on the heat kernel so that the solutions of the associated heat equations attain the initial data pointwise. We demonstrate that these conditions are satisfied by a broad class of operators, including the Laplace operators perturbed by a gradient, fractional Laplacian, mixed local-nonlocal operators, Laplacian on Riemannian manifolds, Dunkl Laplacian and many more. In addition, we consider the Laplace operator in $\mathbb{R}^n$ with the Hardy potential and establish a characterization for the pointwise convergence to the initial data. We also prove similar results for the nonhomogeneous equations and showcase an application for the power-type nonlinearities.